# Understanding the portfolio used in the derivation of the Black-Scholes PDE

In the derivation of the Black-Scholes equation, I see that a portfolio $$\Pi=V-\Delta S$$ is used, where $$\Delta$$ turns out to be $$\frac{\partial{V}}{\partial S}$$. But to determine $$\Delta$$, it is assumed to be constant. So my confusion comes from the reasoning why $$\Delta$$ is constant as when I look at the graphed solutions for a call option, $$\frac{\partial{C}}{\partial S}$$ is clearly not constant when S is less than the price of the strike price.

• Is your $y$-axis $C$, $V$ or $\Pi$? It may be worth editing in a proof that $\partial_S V$ being constant would imply $\partial_S C$ is too. – J.G. Dec 27 '18 at 17:51
• @J.G. the y-axis is C- the call option value and in derivations, people state $d\Pi=dV-\Delta dS$ implying |delta is a constant. – DLB Dec 27 '18 at 19:37